Digitized images frequently require large amounts of data. Medical imaging, for instance, often produces huge volumes of data to be stored or communicated to another location. Personal-computer disk drives increasingly store photographs and scenics for realistic games. Burgeoning security applications promise to require increasing amounts of image data for biometric scanning. Transmitting only part of the data from an image would substantially reduce bandwidth and time requirements for communicating images or other large data sequences, and would slash storage volume for recording such images.
Moreover, images might be produced only as a magnitude or power component of frequency transformed data. Some astronomical and weather data fall into this category. Other applications may severely distort the phase or magnitude component of a transformed image, leaving reliable data only in the complementary magnitude or phase component. Blind convolution, for example, attempts to restore data from an unknown signal convolved with noise or filtered through an unknown system.
Using the Fourier transform of a spatial image as a paradigm, the magnitude component of the transform contains half the information of the image from the spatial base domain, and the complementary phase component contains the other half. The ability to restore or reconstruct an image from only one of these frequency components could save the bandwidth and/or time required to transmit both components, or to transmit the image in its spatial or other base-domain form. Storing only a single transform component would also substantially reduce the space required to hold a representation of a transform-domain image on a disk or other medium. These reductions are especially significant in the normal case where transform-domain data requires a double-precision floating-point format.
However, those in the art currently regard adequate restoration or reconstruction from only one component of the transform domain as an open problem. Techniques do exist for certain specialized cases or for images having particular constraints. Some known techniques require initial estimates close to the final image. Others lead to non-unique solutions. When the conditions fail, conventional iterative procedures often wander about the solution space without converging to a unique answer. Sometimes even determining whether or not the proper conditions obtain is difficult.